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  2. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P ( n ) represent " 2 n − 1 is odd": (i) For n = 1 , 2 n − 1 = 2(1) − 1 = 1 , and 1 is odd, since it leaves a remainder of 1 when divided by 2 .

  3. List of incomplete proofs - Wikipedia

    en.wikipedia.org/wiki/List_of_incomplete_proofs

    In 1925 Ackermann published a proof that a weak system can prove the consistency of a version of analysis, but von Neumann found an explicit mistake in it a few years later. Gödel's incompleteness theorems showed that it is not possible to prove the consistency of analysis using weaker systems. Groups of order 64.

  4. Convergence proof techniques - Wikipedia

    en.wikipedia.org/wiki/Convergence_proof_techniques

    Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.

  5. Algebraic expression - Wikipedia

    en.wikipedia.org/wiki/Algebraic_expression

    In mathematics, an algebraic expression is an expression built up from constants (usually, algebraic numbers) variables, and the basic algebraic operations: addition (+), subtraction (-), multiplication (×), division (÷), whole number powers, and roots (fractional powers).

  6. Algebraic function - Wikipedia

    en.wikipedia.org/wiki/Algebraic_function

    In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation.Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power.

  7. Noether normalization lemma - Wikipedia

    en.wikipedia.org/wiki/Noether_normalization_lemma

    Theorem. (Noether Normalization Lemma) Let k be a field and = [′,..., ′] be a finitely generated k-algebra.Then for some integer d, , there exist , …, algebraically independent over k such that A is finite (i.e., finitely generated as a module) over [, …,] (the integer d is then equal to the Krull dimension of A).

  8. Algebraic geometry and analytic geometry - Wikipedia

    en.wikipedia.org/wiki/Algebraic_geometry_and...

    In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables.

  9. Algebraic independence - Wikipedia

    en.wikipedia.org/wiki/Algebraic_independence

    In abstract algebra, a subset of a field is algebraically independent over a subfield if the elements of do not satisfy any non-trivial polynomial equation with coefficients in . In particular, a one element set { α } {\displaystyle \{\alpha \}} is algebraically independent over K {\displaystyle K} if and only if α {\displaystyle \alpha } is ...